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APPENDIX III
Derivation of the rate equation for the two kinetic effect observed in the reaction between gold(III) and PADA in DTAC
The species prevailing in water solution at acidic pH is AuCl4−. However, our experiments concerning the dependence of the reaction rate on [Cl−] suggest that in DTAC at pH values less than 2 the reactive species is AuCl3(H2O) which is in equilibrium with AuCl4−.
Fast effect
The most reasonable reaction scheme which, among the various schemes tested, can rationalize the kinetic results provided by the analysis of the fast effect, observed for pH values less than 2.5, is the one depicted below.
Consider the reaction scheme (III.1)
(III.1)
The mass concentration with respect to gold reads
CAu = [AuCl4−] + [AuCl3(H2O)] + [AuCl3L-LH+] (III.2)
The equilibrium constants of the equilibrium step is
] Cl )][ O H ( AuCl [ ] AuCl [ K − − = 2 3 4 (III.3) Cl− +AuCl3(H2O) AuCl4− + H 2O K AuCl3(H2O) + L- LH+ AuClk2 3L-LH+ + H2O k-2
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Taking into account that in equation (III.2) [AuCl3L] can be neglected compared with the other species since CAu >> CL, introduction of equation (III.3) in equation (III.2) yields equation (III.4).
] Cl [ K 1 C )] O H ( AuCl [ 3 2 Au − + = (III.4)
The mass conservation equation for PADA is
CPADA = [H2L2+] + [HL+] + [L] + [AuCl3L-LH+] (III.5)
Where, for simplicity, L denotes the bidentate L-L PADA ligand, HL+ is L-LH+ and soon. Its differential form is
−δ[AuCl3L-LH+] = δ[H2L2+] + δ[HL+] + δ[L] ≡δ[Lf] (III.6) where δ[Lf] denotes the total concentration shift of the ligand from equilibrium.
The acid dissociation constants of PADA can be expressed in differential form
] L H [ ] HL [ ] H [ KA + + + = δδ 2 2 1 (III.7) and ] HL [ ] L [ ] H [ KA + + δ δ = 2 (III.8)
Introduction of equations (III.7) and (III.8) in equation (III.9) yields
+ + + α δ = δ = − δ − HL f 3 ] HL [ ] L [ ] LH L AuCl [ (III.9)
123 where 2 A 1 A 1 A 2 1 A HL [H ] K [H ] K K ] H [ K + + = α + + ++ (III.10)
The rate of appearance of AuCl3L-LH+ is written in differential form as
] LH L AuCl [ k ] HL [ )] O H ( AuCl [ k dt ] LH L AuCl [ d 3 2 2 3 2 3 + − + + − δ − δ = − δ (III.11)
Introduction of equations (III.4) and (III.9) into equation (III.11), yields equation (III.12). ] LH L AuCl [ k ] Cl [ K 1 C k dt ] LH L AuCl [ d 3 2 HL Au 2 3 + − − + + − δ + + α = − δ (III.12)
Variable separation and integration of equation (III.12) yields the expression of 1/τf as a function of the reactant concentration.
2 HL Au 2 f k ] Cl [ K 1 C k 1 − − + + + α = τ (III.13)
According to equation (III.13) the time constant 1/τf increases linearly with the total gold(III) concentration while displays a decrease on rising [Cl−], as shown experimentally. The pH dependence of the reaction is included in the variable αHL+. Equation (III.10) shows that αHL+ displays a maximum in the investigated range of concentration, in agreement with the experiments.
The slow effect
The reaction scheme proposed is the following
(III.14) (III.15) Au(OH)Cl3−+ L L AuOHCl+ L L k1 k-1 + 2Cl− Au(OH)2Cl2− +L L Au(OH)2+ L L k2 k-2 + 2Cl−
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(III.16)
(III.17)
The mass conservation equation for gold between pH 4 and 7 is
CAu = [AuOHCl3−]+[Au(OH)2Cl2−] (III.18)
Having neglected the species buond to [L] since these are minority. Steps (III.16) and (III.17) are fast and their equilibrium constants are
] Cl ][ Cl ) OH ( Au [ ] OH ][ AuOHCl [ K 2 2 3 − − − − = (III.19) ] Cl ][ LL ) OH ( Au [ ] OH ][ ClLL ) OH ( Au [ K 2 + − − + = ′ (III.20)
Introduction of equation (II.19) in (II.18) yields
] Cl [ K ] OH [ ] OH [ C ] Cl ) OH ( Au [ 2 2 Au − − − − + = (III.21) and ] Cl [ K ] OH [ ] Cl [ K C ] Cl ) OH ( Au [ 3 A−u − − − + = (III.22) The mass balance equation for L L, expressed in differential form is
(III.23)
Introduction of equation (III.20) in its differential form into equation (III.23) gives Au(OH)2Cl2− + Cl− K Au(OH)Cl3− + OH− Au(OH)2+ L L K' + OH− AuOHCl+ L L + Cl− L L + δ[Au(OH)2+ ] δ[L-L]+δ[Au(OH)Cl+ ]L L = 0
125 (III.24)
The rate law, in differential form, is :
(III.25)
Introductions of equations (III.19), (III.21), (III.22), and (III.24) into (III.25) yields
(III.26)
Introduction of equation (II.24) in III.26 gives
L] δ[L ] K[Cl ] [OH ] [OH ] [Cl k ] [OH ] [Cl K k C ] K[Cl ] [OH ] [OH k ] K[Cl ] [OH ] K[Cl k dt L] dδδ[ 2 2 1 Au 2 1 − + + ′ + + + + = − − − − − − − − − − − − − − − − (III.27)
Variable separation and integration of equation (III.27) provides the expression for 1/τs in the form of equation (II.28) which corresponds to equation (5.5) of the text ] Cl [ K ] OH [ ] OH [ ] Cl [ k ] OH [ ] Cl [ K k C ] Cl [ K ] OH [ ] OH [ k ] Cl [ K ] OH [ ] Cl [ K k / 1 2 2 1 Au 2 1 s − − − − − − − − − − − − − − ′ + + ′ + + + + = τ (III.28) L L δ[Au(OH)2+ ] -δ[L-L]= L L k-1δ[Au(OH)Cl+ ]+ k-2δ[Au(OH)2+ ] [ClLL -]2 δ[Au(OH)2+ ]L L
